First, I apologize to Reuben Hersh for misspelling his name. The
mistake was inadvertent.
Secondly, I thought it might be useful (especially for
non-philosophers on this list) to explain some of the basics about
conceptual analysis and thought experiments, which trained
philosophers take for granted, and which they use as a matter of
second nature in metaphysical, epistemological, and metaethical
argumentation. It strikes me from some of Hersh's responses (to my
earlier questions, challenges or arguments) that he might not be
familiar with some of the argumentative techniques of contemporary
analytical philosophy. This is not meant in any way as a personal
criticism, since members of this list have widely varying kinds of
disciplinary expertise. But some brief words on these argumentative
techniques might help communication generally among members of the
list, some of whom do, and some of who do not, have the kind of
expertise in analytical argumentation that I would like to explain.
Once the explanation is given, I shall then point out how Hersh has
failed to engage my objections to his philosophical doctrine about the
nature of numbers.
Philosophers who engage in conceptual analysis are concerned with the
*modal* features of relationships among concepts. Thus we find them
considering such questions as whether it is *necessary* to have a
language in order to have thoughts; or whether it is *necessary* to be
embodied in order to be a locus of consciousness; or whether it is
*possible* that there be a true proposition that is in principle
*unknowable*. A metaphysician might inquire whether any of one's
properties is *essential* to one's identity. An epistemologist might
inquire whether *there could be* a justified true belief that did not
amount to knowledge. And a moral philosopher might inquire whether the
moral qualities of an act *necessarily involve* the agent's intentions.
In all these examples, the modal element is crucial. Words like
`necessary', `possible', and `essential' are clues to the
philosophical nature of the concern. Note that even modern
philosophers like Quine and Davidson, who are opposed to any purported
distinction between truths of fact and truths of meaning, nevertheless
display considerable expertise as *analytical* philosophers in the way
they treat the *conceptual* issues posed by questions with the modal
flavour of the examples just given.
Philosophy importantly involves (among other things) pursuing
arguments for conclusions about the `fundamental' or `ultimate' nature
of: consciousness; rationality; morality; freewill v. determinism;
concrete v. abstract existence; mathematical knowledge; scientific
knowledge; space and time; concepts, properties, universals; etc. As a
reflective and `a priori' enterprise, it is often forced to rest its
arguments on premisses or assumptions that can be justified by
introspection, or by appeals to intuition.
Philosophical reasoning (like mathematical reasoning) can be
criticized in either one of two ways: (1) one can object to the
assumed truth of a basic assumption made in the course of the
argument, and taken for granted for the purposes of that argument; or
(2) one can object to the validity or suasive power of a particular
step or transition made in the course of the argument.
In meeting a criticism of the first kind, the proponent of the
argument has to beware of the dangers of (a) infinite regress, (b)
circularity, and (c) inconsistency or incoherence of the set of
premisses eventually taken as 'ultimate'.
In mathematics, criticisms of the second kind (of fallacies in
deductive reasoning) can be backed up by displaying a
`counterexample'. This will be some model or construction that
satisfies the immediate premisses for the questionable step, but
falsifies its conclusion. As a teacher of mathematics, Hersh will no
doubt have considerable experience doing this for his students. If a
student, for example, makes a step from `f is continuous on (0,1)' to
`f is uniformly continuous on (0,1)', Hersh will no doubt say
something like `No, no! Consider the function f(x) = 1/x. This is
continuous on (0,1) but, because it gets ever steeper as x tends to 0,
one's choice of delta to take care of every choice of tolerance
epsilon is going to depend on the point x in question ...'.
So too in philosophy. When someone draws a questionable (i.e.
fallacious) inference in philosophical argumentation, the way we
criticise it effectively is to engage in a thought experiment---one
which reveals that the premisses *could be true* while yet the
conclusion *would turn out false*. The literature of contemporary
analytical philosophy is replete with superbly inventive, ingenious
and creative thought experiments of this kind. (In fact, it sometimes
seems to me that philosophers acquire greater reputations for their
ability to construct counterexamples than for their ability to
prosecute long trains of valid reasoning!)
Many an important philosophical project takes the form of trying to
state necessary and sufficient conditions for the application of some
philosophically important notion; or at least 'explicating' the
notion, or effecting some kind of worthwhile philosophical reduction
of it to notions that are more familiar, or about whose application we
can be more certain. Among the many famous examples it will suffice to
mention four.
(i) Tarski gave an explication of the truth-conditions of sentences of
extensional languages in terms of the reference of the primitive
extralogical terms occurring in those sentences.
(ii) Grice gave an analysis of what it is for a speaker to mean that p
by an utterance, an analysis that appeals to the speaker's intentions
regarding his/her audience's beliefs.
(iii) Lewis gave an analysis of conventions (especially linguistic
ones) in terms of the reciprocal beliefs and expectations concerning
other people's coordinated behaviour when they face a recurring
communal problem, and are aware of past salient solutions.
(iv) The fourth example is of longer standing but of more dubious
success: the attempt to analyse or explicate the notion of cognitive
significance, so as to reveal just how the empirical content of
theoretical sentences is acquired from the logical relationships they
bear, within the scientific theory in question, to the observation
sentences by means of which the theory can be tested. This tradition
began with Ayer, continued through Carnap, and petered out with
Hempel.
When analyses like these are offered, the race is on to find
counterexamples that will force a refinement of the analysis, or even
the eventual abandonment of the analytical project in question. This
is when philosophical energies become focused on the search for
ingenious counterexamples. One tries to set up a thought experiment in
which the analysans would be true, but the analysandum false, or vice
versa.
Someone outside academic philosophical circles may not be aware of
just what a staple the far-fetched thought experiment has become.
In no particular order, here are some of the thought experiments that
are a standard part of the analytical philosopher's intellectual
tool-kit. The purpose of each is mentioned in brackets.
0) (perhaps the progenitor of them all?:) Descartes' evil demon, who
is supposed to be able systematically to deceive Descartes in all his
thinking about the external world [thereby to cast doubt on any
inference roughly like "I (appear to myself directly to) perceive that
p, and that perceptual impression coheres with all my other
impressions; therefore, it is true that p"]
1) Putnam's scenario of the `brain in a vat' whose sensory
stimulations are supplied by a mad scientist, and whose afferent nerve
signals are monitored by said scientist [to make the problem of
scepticism about the external world more compelling];
2) Putnam's thought experiment involving `Twin Earth', where everyone
and everything on Earth has a molecule-for-molecule Doppelg"anger,
except that what appears to be water on Twin Earth has the chemical
constitution XYZ rather than H2O [to challenge the assumption that
`meanings are in the head'];
3) Gettier's example of the fellow who has a red Ford, which gets
stolen but is promptly replaced with an exact replica [to challenge
the claim that knowledge is justified true belief];
4) Williams's example of the two people who are offered the prospect
of the `character and memory switch', followed by reward to one and
punishment to the other [to raise doubts about whether bodily
continuity is sufficient for personal identity];
5) Modern resuscitation of Locke's problem of the inverted spectrum
(where, say, one's impressions of red and green get switched) [to
challenge the functionalist in the philosophy of mind, who claims that
every mental state is characterized by the logical role it plays as a
state in the `program' that `is' the human mind];
6) Jackson's example of the scientist Mary, who knows everything there
is to know about neurochemistry etc., but who sees everything in
black, white and greys [to challenge the notion that materialism could
hold all the truth there is to know about mental experience];
7) Kripke's example of the standard meter rod in Paris [to challenge
the inference from `proposition P is known a priori' to `proposition P
is necessary'];
8) Goodman's example of the predicate `x is grue', (i.e. x is examined
before 2,000 and is green, or x is not so examined and is blue) [to
challenge the inductivist who believes that one can project any
well-defined property holding of a large sample of natural things to
all members of the relevant population];
9) Ayer's example of the universe consisting of five sounds [in order
to challenge the assumption that time has to be linear];
10) Searle's example of the monoglot Englishman in the `Chinese room',
using a manual (written in English) to produce responses, written in
Chinese characters, to inputs that are also written in Chinese
characters [to challenge the notion that all that is involved in
linguistic understanding is mechanical, `symbolic' computation];
11) The famous example of the ship of Theseus (which is re-built plank
by plank, the removed planks then being reassembled to make another
(?) or the original (?) ship);
12) Block's example of every person in China having a walkie-talkie
radio, so that the whole population is like a neural network,
instantiating a functionalist's program of supposedly mental states;
13) The favourite example wielded against behaviourists in the
philosophy of mind: *zombies*, namely creatures that are constituted,
and behave, like human beings, but who (if a personal pronoun is
appropriate here), we are invited to imagine, have no inner
experiences at all;
14) Cases involving 'deviant causal chains' [to create problems for
any attempt to give a causal analysis of knowledge];
15) Runaway railroad car examples, and doomsday machine examples [to
raise problems for various utilitarian accounts of what the right
thing is to do];
16) Massey's Cretan translation manual [involved in an attempt to give
a clinching example of the necessarily indeterminate character of
translation];
17) Telecloning and split-brain examples [to create problems for
various accounts of personal identity];
18) Kripke's quadder (who interprets 'x+y' in such a way as to get the
sum of x and y if they are both less than 57, but to get 5 otherwise)
[to create problems for the view that what a speaker means by a
certain expression must be determined by past and present facts
concerning the speaker];
19) Robinson Crusoe examples [to challenge the view that thought and
language are necessarily social;
... and many, many more.
Almost all of these thought experiments involve absurdly improbable
scenarios, but that does not detract at all from their dialectical
potency. When students learn philosophy they learn, among other
things, how to open their minds to the most far-fetched, bizarre
possibilities---situations which stretch certain concepts to their
limits, in order to explore the conceptual connections among the
concepts themselves. In acquiring this intellectual skill, they have
to learn to put aside their beliefs concerning the *probability* or
*likelihood* or *feasibility* of the imagined scenarios. Convictions
about how remote these possibilities may be, given one's everyday
experience (or even given one's scientific knowledge as well) will
actually *detract* from the student's ability to gain increased
*philosophical* understanding of what is at issue. All that is
necessary is that the thought experiment should be conducted in such
clear and simple terms that certain `truths within it' will be granted
intuitively, or upon appropriate reflection. Skilfully arranged, the
right scenario will force membership in such a set of truths upon the
premisses of the challenged inference, and upon the negation of its
conclusion.
It should by now be clear that, in my exchange with Hersh, I was
hoping to draw him into serious reflection upon his views about the
allegedly non-abstract and humanity-dependent nature of numbers, by
eliciting certain conclusions from those views, and then constructing
(perfectly licitly) certain thought experiments that would confound
those conclusions. One of my thought experiments involved inviting
Hersh to consider the *conceptual possibility* of extraterrestrial
intelligent beings who are capable of mathematical thought and its
linguistic communication. (It was not even required that I maintain
such ETI as a *physical* possibility modulo our current best physical
theories.) Another of my thought experiments invited Hersh to consider
the possible extinction of all intelligent beings, followed by their
serendipitous evolutionary recapitulation, to the point of
re-developing number theory in the same form that it had with their
cosmic precursors. The purpose of the thought experiments was to
create difficulty for the view that the natural numbers would cease to
exist if human beings ceased to exist; and for the view that human
intellects would have a total monopoly on intellectual access to the
natural numbers. Both of these views were explicit, or clearly
implicit, in what Hersh had maintained.
Hersh's disbelief in the likelihood of such beings or sequences of
events would be *entirely beside the point* in such a conceptual
exercise. (And he can rest assured that this is not another
"astonishing dictum"!) In fact, he ought to be informed that, among
professional philosophers at least, a certain erudite delight is
sometimes to be had from making a well-known thought experiment *even
more preposterous and far-fetched* than the original, in order to
establish yet another conceptual point not made by the original. The
perverse intellectual delight stems from making the needed
`intuitively evident' truths in the weird situation intuitively
evident *despite* the utterly far-fetched nature of the imagined
scenario!
A final observation on the method of the thought experiment: it is
customary, also, to be mindful of the distinctions between the
following kinds of possibility (and the respective dual notions of
necessity):
epistemic; physical; metaphysical; conceptual; logical.
There is also the problem of where "the imaginable" fits in with
respect to these various kinds of possibilities. I shall not go into
the niceties involved here, unless it transpires that there is
sufficient demand for further clarification of philosophical method.
Allow me one more thought in closing, to bring philosophical
argumentation back into contact with mathematical argumentation. The
famous Cambridge mathematician Littlewood related a story of his visit
to his old high school, to give an invited talk to the sixth form (=
senior year, for those in the USA). He was giving them a simple proof,
in the course of which he said "Suppose now that x is a real number."
A hand went up in the back of the room. "But Sir, suppose x ISN'T a
real number!"
Clearly the young lad had failed to grasp what was going on with "let
clauses" in the kinds of quantified reasoning so essential to modern
mathematics. These forms of reasoning are now perfectly well
understood, and are codified exactly in highly developed systems of
natural deduction.
In the same way, anyone who scoffs at a thought experiment on the
grounds that the envisaged situation is hugely unlikely, or irrelevant
to daily human concerns, has simply failed to grasp what is going on
with the "Let's imagine a case where..." clauses in the kinds of
thought-experimental reasoning so essential to contemporary
philosophy. These forms of reasoning are perfectly well understood,
and are a core component of highly developed forms of philosophical
argumentation. Naturally, being more discursive and open-ended, they
have not been codified in the precise way that first-order logical
reasoning has been. But that is not the crucial issue at stake in this
analogy.
Neil Tennant